Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,12)(9,21)(10,14)(11,18)(13,24)(16,32)(17,29)(19,27)(20,30)(22,25)(23,28)(26,34)(31,33)(35,36);(1,2)(3,4)(8,19)(11,25)(12,27)(15,29)(20,23)(26,36)(28,30)(31,32);(5,7)(6,11)(8,16)(9,22)(10,17)(12,26)(13,29)(14,25)(15,21)(18,24)(19,33)(20,35)(23,34)(27,36)(28,31)(30,32);(2,4)(5,8)(6,12)(7,16)(9,23)(10,19)(11,26)(13,30)(14,27)(15,31)(17,33)(18,35)(20,24)(21,28)(22,34)(25,36)(29,32)Polycyclic presentation:
2^9|f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285083, #72285600, #72285668, … |
| maximal normal subgroup of | #72285083, #72285600, #72285668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324737);